Question: The sum of the first 15 terms of a geometric sequence with 7 as the first term and a common ratio of -3 is ____________.
1. **State the problem:** We need to find the sum of the first 15 terms of a geometric sequence where the first term $a = 7$ and the common ratio $r = -3$.
2. **Formula for the sum of the first $n$ terms of a geometric sequence:**
$$S_n = a \frac{1 - r^n}{1 - r}$$
where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.
3. **Apply the formula:**
$$S_{15} = 7 \frac{1 - (-3)^{15}}{1 - (-3)}$$
4. **Calculate the denominator:**
$$1 - (-3) = 1 + 3 = 4$$
5. **Calculate the numerator:**
First, calculate $(-3)^{15}$.
Since $(-3)^{15} = -(3^{15})$ because the exponent is odd.
Calculate $3^{15}$:
$$3^{15} = 14348907$$
So,
$$(-3)^{15} = -14348907$$
6. **Substitute back into numerator:**
$$1 - (-14348907) = 1 + 14348907 = 14348908$$
7. **Calculate the sum:**
$$S_{15} = 7 \times \frac{14348908}{4}$$
8. **Simplify the fraction:**
$$\frac{14348908}{4} = 3587227$$
9. **Multiply:**
$$7 \times 3587227 = 25110589$$
**Final answer:**
$$S_{15} = 25110589$$
This corresponds to option b.) 25,110,589.